Algebra Examples

$h (x) = 2 x^{3} + 3$

Step 1

Write $h (x) = 2 x^{3} + 3$ as an equation.

$y = 2 x^{3} + 3$

Step 2

Interchange the variables.

$x = 2 y^{3} + 3$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $2 y^{3} + 3 = x$ .

$2 y^{3} + 3 = x$

Step 3.2

Subtract $3$ from both sides of the equation.

$2 y^{3} = x - 3$

Step 3.3

Divide each term in $2 y^{3} = x - 3$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 y^{3} = x - 3$ by $2$ .

$\frac{2 y^{3}}{2} = \frac{x}{2} + \frac{- 3}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 y^{3}}{2} = \frac{x}{2} + \frac{- 3}{2}$

Step 3.3.2.1.2

Divide $y^{3}$ by $1$ .

$y^{3} = \frac{x}{2} + \frac{- 3}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$y^{3} = \frac{x}{2} - \frac{3}{2}$

Step 3.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{\frac{x}{2} - \frac{3}{2}}$

Step 3.5

Simplify $\sqrt[3]{\frac{x}{2} - \frac{3}{2}}$ .

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Step 3.5.1

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{x - 3}{2}}$

Step 3.5.2

Rewrite $\sqrt[3]{\frac{x - 3}{2}}$ as $\frac{\sqrt[3]{x - 3}}{\sqrt[3]{2}}$ .

$y = \frac{\sqrt[3]{x - 3}}{\sqrt[3]{2}}$

Step 3.5.3

Multiply $\frac{\sqrt[3]{x - 3}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$y = \frac{\sqrt[3]{x - 3}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 3.5.4

Combine and simplify the denominator.

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Step 3.5.4.1

Multiply $\frac{\sqrt[3]{x - 3}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$y = \frac{\sqrt[3]{x - 3} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 3.5.4.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$y = \frac{\sqrt[3]{x - 3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 3.5.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{\sqrt[3]{x - 3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 3.5.4.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{x - 3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 3.5.4.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 3.5.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{x - 3} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 3.5.4.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{\sqrt[3]{x - 3} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 3.5.4.5.3

Combine $\frac{1}{3}$ and $3$ .

$y = \frac{\sqrt[3]{x - 3} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 3.5.4.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{x - 3} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 3.5.4.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{x - 3} {\sqrt[3]{2}}^{2}}{2^{1}}$

Step 3.5.4.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{x - 3} {\sqrt[3]{2}}^{2}}{2}$

Step 3.5.5

Simplify the numerator.

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Step 3.5.5.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$y = \frac{\sqrt[3]{x - 3} \sqrt[3]{2^{2}}}{2}$

Step 3.5.5.2

Raise $2$ to the power of $2$ .

$y = \frac{\sqrt[3]{x - 3} \sqrt[3]{4}}{2}$

Step 3.5.6

Simplify with factoring out.

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Step 3.5.6.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(x - 3) \cdot 4}}{2}$

Step 3.5.6.2

Reorder factors in $\frac{\sqrt[3]{(x - 3) \cdot 4}}{2}$ .

$y = \frac{\sqrt[3]{4 (x - 3)}}{2}$

Step 4

Replace $y$ with $h^{- 1} (x)$ to show the final answer.

$h^{- 1} (x) = \frac{\sqrt[3]{4 (x - 3)}}{2}$

Step 5

Verify if $h^{- 1} (x) = \frac{\sqrt[3]{4 (x - 3)}}{2}$ is the inverse of $h (x) = 2 x^{3} + 3$ .

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Step 5.1

To verify the inverse, check if $h^{- 1} (h (x)) = x$ and $h (h^{- 1} (x)) = x$ .

Step 5.2

Evaluate $h^{- 1} (h (x))$ .

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Step 5.2.1

Set up the composite result function.

$h^{- 1} (h (x))$

Step 5.2.2

Evaluate $h^{- 1} (2 x^{3} + 3)$ by substituting in the value of $h$ into $h^{- 1}$ .

$h^{- 1} (2 x^{3} + 3) = \frac{\sqrt[3]{4 ((2 x^{3} + 3) - 3)}}{2}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Subtract $3$ from $3$ .

$h^{- 1} (2 x^{3} + 3) = \frac{\sqrt[3]{4 (2 x^{3} + 0)}}{2}$

Step 5.2.3.2

Add $2 x^{3}$ and $0$ .

$h^{- 1} (2 x^{3} + 3) = \frac{\sqrt[3]{4 \cdot (2 x^{3})}}{2}$

Step 5.2.3.3

Multiply $4$ by $2$ .

$h^{- 1} (2 x^{3} + 3) = \frac{\sqrt[3]{8 x^{3}}}{2}$

Step 5.2.3.4

Rewrite $8 x^{3}$ as ${(2 x)}^{3}$ .

$h^{- 1} (2 x^{3} + 3) = \frac{\sqrt[3]{{(2 x)}^{3}}}{2}$

Step 5.2.3.5

Pull terms out from under the radical, assuming real numbers.

$h^{- 1} (2 x^{3} + 3) = \frac{2 x}{2}$

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

$h^{- 1} (2 x^{3} + 3) = \frac{2 x}{2}$

Step 5.2.4.2

Divide $x$ by $1$ .

$h^{- 1} (2 x^{3} + 3) = x$

Step 5.3

Evaluate $h (h^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$h (h^{- 1} (x))$

Step 5.3.2

Evaluate $h (\frac{\sqrt[3]{4 (x - 3)}}{2})$ by substituting in the value of $h^{- 1}$ into $h$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 {(\frac{\sqrt[3]{4 (x - 3)}}{2})}^{3} + 3$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the product rule to $\frac{\sqrt[3]{4 (x - 3)}}{2}$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{{\sqrt[3]{4 (x - 3)}}^{3}}{2^{3}}) + 3$

Step 5.3.3.2

Simplify the numerator.

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Step 5.3.3.2.1

Rewrite ${\sqrt[3]{4 (x - 3)}}^{3}$ as $4 (x - 3)$ .

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Step 5.3.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 (x - 3)}$ as ${(4 (x - 3))}^{\frac{1}{3}}$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{{({(4 (x - 3))}^{\frac{1}{3}})}^{3}}{2^{3}}) + 3$

Step 5.3.3.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{{(4 (x - 3))}^{\frac{1}{3} \cdot 3}}{2^{3}}) + 3$

Step 5.3.3.2.1.3

Combine $\frac{1}{3}$ and $3$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{{(4 (x - 3))}^{\frac{3}{3}}}{2^{3}}) + 3$

Step 5.3.3.2.1.4

Cancel the common factor of $3$ .

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Step 5.3.3.2.1.4.1

Cancel the common factor.

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{{(4 (x - 3))}^{\frac{3}{3}}}{2^{3}}) + 3$

Step 5.3.3.2.1.4.2

Rewrite the expression.

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{4 (x - 3)}{2^{3}}) + 3$

Step 5.3.3.2.1.5

Simplify.

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{4 (x - 3)}{2^{3}}) + 3$

Step 5.3.3.2.2

Apply the distributive property.

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{4 x + 4 \cdot - 3}{2^{3}}) + 3$

Step 5.3.3.2.3

Multiply $4$ by $- 3$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{4 x - 12}{2^{3}}) + 3$

Step 5.3.3.2.4

Factor $4$ out of $4 x - 12$ .

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Step 5.3.3.2.4.1

Factor $4$ out of $4 x$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{4 (x) - 12}{2^{3}}) + 3$

Step 5.3.3.2.4.2

Factor $4$ out of $- 12$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{4 x + 4 \cdot - 3}{2^{3}}) + 3$

Step 5.3.3.2.4.3

Factor $4$ out of $4 x + 4 \cdot - 3$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{4 (x - 3)}{2^{3}}) + 3$

Step 5.3.3.3

Raise $2$ to the power of $3$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{4 (x - 3)}{8}) + 3$

Step 5.3.3.4

Cancel the common factor of $2$ .

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Step 5.3.3.4.1

Factor $2$ out of $8$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{4 (x - 3)}{2 (4)}) + 3$

Step 5.3.3.4.2

Cancel the common factor.

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = 2 (\frac{4 (x - 3)}{2 \cdot 4}) + 3$

Step 5.3.3.4.3

Rewrite the expression.

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = \frac{4 (x - 3)}{4} + 3$

Step 5.3.3.5

Cancel the common factor of $4$ .

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Step 5.3.3.5.1

Cancel the common factor.

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = \frac{4 (x - 3)}{4} + 3$

Step 5.3.3.5.2

Divide $x - 3$ by $1$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = x - 3 + 3$

Step 5.3.4

Combine the opposite terms in $x - 3 + 3$ .

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Step 5.3.4.1

Add $- 3$ and $3$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$h (\frac{\sqrt[3]{4 (x - 3)}}{2}) = x$

Step 5.4

Since $h^{- 1} (h (x)) = x$ and $h (h^{- 1} (x)) = x$ , then $h^{- 1} (x) = \frac{\sqrt[3]{4 (x - 3)}}{2}$ is the inverse of $h (x) = 2 x^{3} + 3$ .

$h^{- 1} (x) = \frac{\sqrt[3]{4 (x - 3)}}{2}$